Local Geometry of Real and Complex Analytic Mappings

实数和复数解析映射的局部几何

• 批准号: RGPIN-2018-04239
• 负责人: Adamus, Janusz
• 金额: $ 1.68万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758474
• 关键词: Local; Geometry; Real; Complex; Analytic

The proposed research program is concerned with the local geometry of real and complex analytic mappings and their images. It lies at the interface of analytic geometry, commutative algebra, singularity theory and theory of functions of several complex variables, and involves problems in all these directions. The program builds on very successful work done under my present NSERC discovery grant.The two main objectives of the program are:(a) Establishing of a computable classification of singularities of analytic and polynomial mappings.(b) Development of a singular CR geometry.The first of the above long term goals is concerned with the local geometry of mappings from the point of view of singularity theory.Our main idea is to reduce the study of the geometric complexity of a map to an automated calculation of certain algebraic invariants. To achieve this goal, we need to develop new criteria that would characterize the geometry of a map in terms of algebraic properties of certain modules associated with the map, which can be verified by means of computer algebra methods. Examples of such criteria are our recent characterizations of openness and flatness of maps. Our approach is to study degeneracies (or discontinuities) in the family of fibres of a given map. These are often too subtle to be detected on the algebraic level, and so we need to amplify these discontinuities to the extent that they get reflected in algebraic properties of certain modules associated with the map. This can be done, for example, by passing to fibred powers of the map.The study of local invariants classifying the mapping singularities is a well established and active area of research. The novelty of our approach lies in its effectiveness, that is, the emphasis on computability. Our most recent results on finite determinacy of flatness and other local properties prove that this approach may be successful even in the transcendental (i.e., non-polynomial) case.Our second long term goal concerns the application of local analytic geometry to the study of real structures in complex ambient spaces. In the non-singular setting, this area of study is known as CR geometry, which can be viewed as a branch of the classical analysis of functions in several complex variables. In this proposal, we consider singular real analytic (even semianalytic) objects in complex spaces. The importance of this approach lies in the fact that such singular sets appear naturally in complex analytic considerations (e.g., as boundaries of complex domains).Singular analytic sets, of course, are not CR manifolds themselves. However, as we showed recently, they admit a stratification into CR manifolds enjoying some very nice differential and algebro-geometric properties. This discovery forms a firm basis for the development of CR geometry in the singular context, and allows us to use the methods of semialgebraic and semianalytic geometry.

拟议的研究计划是关注的局部几何的真实的和复杂的解析映射及其图像。它位于解析几何，交换代数，奇异性理论和理论的职能，几个复杂的变量，并涉及问题，在所有这些方向。该计划建立在非常成功的工作下，我目前的NSERC发现grant.The两个主要目标的计划是：（一）建立一个可计算的分类奇异的解析和多项式映射。(b)奇异CR几何的发展.上述长期目标中的第一个是从奇异理论的观点研究映射的局部几何.我们的主要思想是把对映射的几何复杂性的研究减少到对某些代数不变量的自动计算.为了实现这一目标，我们需要开发新的标准，将其特征的几何形状的地图在代数性质的某些模块与地图，这可以通过计算机代数方法进行验证。这样的标准的例子是我们最近的开放性和平坦性的地图的特征。我们的方法是研究退化（或不连续）在家庭的纤维的一个给定的地图。这些不连续性往往太微妙，无法在代数层面上检测到，因此我们需要放大这些不连续性，使其反映在与映射相关的某些模块的代数属性中。例如，这可以通过传递映射的幂来实现。对映射奇点进行分类的局部不变量的研究是一个成熟而活跃的研究领域。我们的方法的新奇在于它的有效性，即强调可计算性。我们最近关于平坦性和其他局部性质的有限决定性的结果证明，这种方法即使在超越（即，我们的第二个长期目标是将局部解析几何应用于复杂环境空间中真实的结构的研究。在非奇异设置，这方面的研究被称为CR几何，它可以被视为一个分支的经典分析的功能，在几个复杂的变量。在这个建议中，我们考虑奇异的真实的解析（甚至半解析）对象在复杂的空间。这种方法的重要性在于这样一个事实，即这种奇异集自然出现在复杂的分析考虑（例如，作为复域的边界）。奇异解析集，当然，不是CR流形本身。然而，正如我们最近所展示的，它们允许分层成CR流形，享受一些非常好的微分和代数几何性质。这一发现形成了坚实的基础CR几何的发展在奇异的情况下，并允许我们使用的方法，半代数和半解析几何。